28 is the 2nd perfect number.........6 is the 1st perfect number...............13 is the 6th prime........4 and 6 are the 1st two non primes..........exclude 1 and 0........b/c they are special....for the moment anyways...............0, 1, 2, 3, 4, 5, 6.......................exclude 1 and 0................2, 3, 4, 5, 6..........2, 3, 5.........are prime..........the 1st two non primes..........are 4 and 6............the 1st zero....is at 1/2 + 14.13i..................................1st non prime is 4.........like 14i...............the 2nd perfect number divided by the 1st perfect number........is close to F's 1st constant.............................and 6.28...............the 1st perf. number and the 2nd perfect number................is how 2pi starts out............in 1 d measurement.........it is full circle of the unit circle..........................................28 - 6 = 22...........2 + 2 = 4...........2 * 2 = 4...................only zero is the same in that respect........0 + 0 = 0...........0 * 0 = 0.................................there are 4 basic processes in math......basic math.................addition.......multiplication...................subtraction and division...............................multiplication and division..............are in many respects...............the same process...........for the present discussion anyways.......for simplicity's sake..........they are just reverse process of each other................just like addition and subtraction are........4 basic process but in many respects.............2 processes...................

The first constant[edit]

The first Feigenbaum constant is the limiting ratio of each bifurcation interval to the next between every period doubling, of a one-parameter map

${\displaystyle x_{i+1}=f(x_{i}),}$

where f(x) is a function parameterized by the bifurcation parameter a.
It is given by the limit^[2]

${\displaystyle \delta =\lim _{n\to \infty }{\frac {a_{n-1}-a_{n-2}}{a_{n}-a_{n-1}}}=4.669\,201\,609\,\ldots ,}$

where a_n are discrete values of a at the n-th period doubling.
Here is this number to 30 decimal places (sequence A006890 in the OEIS): δ = 4.669201609102990671853203821578…

Illustration[edit]

Non-linear maps[edit]

To see how this number arises, consider the real one-parameter map

${\displaystyle f(x)=a-x^{2}.}$

Here a is the bifurcation parameter, x is the variable. The values of a for which the period doubles (e.g. the largest value for a with no period-2 orbit, or the largest a with no period-4 orbit), are a₁, a₂ etc. These are tabulated below:^[3]